Bounds on codes from smooth toric threefolds with rank(pic(x)) = 2

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dc.contributorSchenck, Henry K.
dc.creatorKimball, James Lee
dc.date.accessioned2010-01-15T00:07:00Z
dc.date.accessioned2010-01-16T01:10:46Z
dc.date.accessioned2017-04-07T19:55:47Z
dc.date.available2010-01-15T00:07:00Z
dc.date.available2010-01-16T01:10:46Z
dc.date.available2017-04-07T19:55:47Z
dc.date.created2008-08
dc.date.issued2009-05-15
dc.description.abstractIn 1998, J. P. Hansen introduced the construction of an error-correcting code over a finite field Fq from a convex integral polytope in R2. Given a polytope P ? R2, there is an associated toric variety XP , and Hansen used the cohomology and intersection theory of divisors on XP to determine explicit formulas for the dimension and minimum distance of the associated toric code CP . We begin by reviewing the basics of algebraic coding theory and toric varieties and discuss how these areas intertwine with discrete geometry. Our first results characterize certain polygons that generate and do not generate maximum distance separable (MDS) codes and Almost-MDS codes. In 2006, Little and Schenck gave formulas for the minimum distance of certain toric codes corresponding to smooth toric surfaces with rank(Pic(X)) = 2 and rank(Pic(X)) = 3. Additionally, they gave upper and lower bounds on the minimum distance of an arbitrary toric code CP by finding a subpolygon of P with a maximal, nontrivial Minkowski sum decomposition. Following this example, we give explicit formulas for the minimum distance of toric codes associated with two families of smooth toric threefolds with rank(Pic(X)) = 2, characterized by G. Ewald and A. Schmeinck in 1993. Lastly, we give explicit formulas for the dimension of a toric code generated from a Minkowski sum of a finite number of polytopes in R2 and R3 and a lower bound for the minimum distance.
dc.identifier.urihttp://hdl.handle.net/1969.1/ETD-TAMU-2992
dc.language.isoen_US
dc.subjecttoric variety
dc.subjectcoding theory
dc.titleBounds on codes from smooth toric threefolds with rank(pic(x)) = 2
dc.typeBook
dc.typeThesis

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